There is software that can record from multiple cameras but I'm not sure how it works with web cams vs. normal surveillance cameras or IP cameras.

Which computer you use would really depend on the software and how the video is being recorded or encoded. The first PC may not have enough CPU or RAM to process 4 streams at one time. If you were just taking still shots then it shouldn't matter.

I just got an idea, but I don't know if it would work. If I would get 4 different loogitech web cams logitechs software would recognize them all but not at once (I guess). So then I could use that software to record but the real question is that would it work, then I should have 4 copies of the software installed.